 Solution

For a function smooth on finitely many closed pieces with finite one-sided derivatives, integration by parts gives its Fourier series coefficients, for $n\ne 0$,where $J_r=f(x_r+)-f(x_r-)$ includes the jump of the periodic extension at the endpoint when necessary. In particular $|c_n|\le C/|n|$. Keeping frequencies $|n|\le K$, with $N\asymp K$, gives squared error $O({\sum }_{n>K}{n}^{-2})=O(N^{-1})$.

A best N-term approximation cannot improve this worst-case rate. For example, $f={\chi }_{[0,1/2)}$ has $|c_n|=1/(\pi |n|)$ on odd nonzero $n$ and zero on even nonzero $n$. Even the optimal selection therefore leaves a squared tail comparable to $N^{-1}$. Thus for the class with jumps,or $N^{-1/2}$ in the $L^2$ norm. Individual functions with no jumps, or additional cancellation, can converge faster. “Smooth except at finitely many points” must include controlled one-sided smoothness; smoothness merely on open pieces permits pathological behavior near their endpoints.